biproductComparison'_comp_biproductComparison

English

The inverse of biproductComparison composed with the mapped first projection returns the original first projection.

Русский

Обратная сторона biproductComparison, композиция сmapped fst-projection возвращает исходную fst-проекцию.

LaTeX

$$$(\mathrm{biproductComparison} F f)^{-1} \gg F.map(\text{prod.fst}) = \text{prod.fst}$$$

Lean4

/-- The composition in the opposite direction is equal to the identity if and only if `F` preserves
the biproduct, see `preservesBiproduct_of_monoBiproductComparison`. -/
@[reassoc (attr := simp)]
theorem biproductComparison'_comp_biproductComparison :
    biproductComparison' F f ≫ biproductComparison F f = 𝟙 (⨁ F.obj ∘ f) := by
  classical
  ext
  simp [biproduct.ι_π, ← Functor.map_comp, eqToHom_map]

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